The history of trigonometry goes back to the earliest recorded mathematics in Egypt and Babylon.
The Babylonians established the measurement of angles in degrees, minutes, and seconds.
Not until the time of the Greeks, however, did any considerable amount of trigonometry exist. In the 2nd century BC the astronomer Hipparchus compiled a trigonometric table for solving triangles.
Starting with 71° and going up to 180° by steps of 71°, the table gave for each angle the length of the chord subtending that angle in a circle of a fixed radius r.
Such a table is equivalent to a sine table. The value that Hipparchus used for r is not certain, but 300 years later the astronomer Ptolemy used r = 60 because the Hellenistic Greeks had adopted the Babylonian base-60 (sexagesimal) numeration system.
In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for steps of 1°, from 0° to 180°, that is accurate to 1/3600 of a unit.
He also explained his method for constructing his table of chords, and in the course of the book he gave many examples of how to use the table to find unknown parts of triangles from known parts. Ptolemy provided what is now known as Menelaus's theorem for solving spherical triangles, as well, and for several centuries his trigonometry was the primary introduction to the subject for any astronomer.
At perhaps the same time as Ptolemy, however, Indian astronomers had developed a trigonometric system based on the sine function rather than the chord function of the Greeks. This sine function, unlike the modern one, was not a ratio but simply the length of the side opposite the angle in a right triangle of fixed hypotenuse. The Indians used various values for the hypotenuse.
Late in the 8th century, Muslim astronomers inherited both the Greek and the Indian traditions, but they seem to have preferred the sine function. By the end of the 10th century they had completed the sine and the five other functions and had discovered and proved several basic theorems of trigonometry for both plane and spherical triangles.
Several mathematicians suggested using r = 1 instead of r = 60; this exactly produces the modern values of the trigonometric functions.
The Muslims also introduced the polar triangle for spherical triangles.
All of these discoveries were applied both for astronomical purposes and as an aid in astronomical time-keeping and in finding the direction of Mecca for the five daily prayers required by Muslim law. Muslim scientists also produced tables of great precision. For example, their tables of the sine and tangent, constructed for steps of 1/60 of a degree, were accurate for better than one part in 700 million.
Finally, the great astronomer Nasir ad-Din at- Tusi wrote the Book of the Transversal Figure, which was the first treatment of plane and spherical trigonometry as independent mathematical Science.
The Latin West became acquainted with Muslim trigonometry through translations of Arabic astronomy handbooks, beginning in the 12th century. The first major Western work on the subject was written by the German astronomer and mathematician Johann Müller, known as Regiomontanus.
In the next century the German astronomer Georges Joachim, known as Rheticus introduced the modern conception of trigonometric functions as ratios instead of as the lengths of certain lines.
The French mathematician François Viète introduced the polar triangle into spherical trigonometry, and stated the multiple-angle formulas for sin(nq) and cos(nq) in terms of the powers of sin(q) and cos(q).

Trigonometric calculations were greatly aided by the Scottish mathematician John Napier, who invented logarithms early in the 17th century. He also invented some memory aids for ten laws for solving spherical triangles, and some proportions (called Napier's analogies) for solving oblique spherical triangles.
Almost exactly one half century after Napier's publication of his logarithms, Isaac Newton invented the differential and integral calculus. One of the foundations of this work was Newton's representation of many functions as infinite series in the powers of x.

Thus Newton found the series sin(x) and similar series for cos(x) and tan(x). With the invention of calculus, the trigonometric functions were taken over into analysis, where they still play important roles in both pure and applied mathematics.
Finally, in the 18th century the Swiss mathematician Leonhard Euler defined the trigonometric functions in terms of complex numbers (see Number). This made the whole subject of trigonometry just one of the many applications of complex numbers, and showed that the basic laws of trigonometry were simply consequences of the arithmetic of these numbers.

from http://www.cartage.org.lb/en/themes/Sciences/Mathematics/Trigonometry/history/History%20.html [Broken]

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